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arXiv · 2607.20387

Godbersen's conjecture and the $L_p$-Rogers-Shephard inequality

Jan Kotrbatý·Mohamed A. Mouamine
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Abstract

We prove that the mixed volume of a convex body with its reflection about the origin is maximized by simplices. This confirms a conjecture of C. Godbersen from 1938 and refines the Rogers-Shephard inequality. We also prove that, among convex polytopes, simplices are the only extremizers. Finally, we use this inequality to prove the $L_p$-version of the Rogers-Shephard inequality for convex bodies containing the origin and show that, for any $p\in(1,\infty]$, the only extremizers are simplices with a vertex at the origin.

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Jan Kotrbatý, Mohamed A. Mouamine. 2026-07-22. Godbersen's conjecture and the $L_p$-Rogers-Shephard inequality. https://arxiv.org/abs/2607.20387

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